Electronic Structure and Symmetryin Periodic Structures

Volker Staemmler

Lehrstuhl für Theoretische Chemie

Ruhr-Universität Bochum

44780 Bochum, Germany

Mariapfarr, February 2016

Outline

I. Introduction

II. Bravais lattices

III. Band structure theory

IV. Space groups

V. Symmetry properties of crystal orbitals and vibrations

VI. Group-subgroup relationships

VII. References

Outline

I. Introduction

II. Bravais lattices

III. Band structure theory

IV. Space groups

V. Symmetry properties of crystal orbitals and vibrations

VI. Group-subgroup relationships

VII. References

Bravais latticesThe fundamental concept for describing 2D and 3D periodic systems is that of a Bravais lattice.

A (2D or 3D) Bravais lattice is defined as an arrangement of lattice points which can be reached from any point of the lattice with displacement (position) vectors of the form

R = n1 a1 + n2 a2 + n3 a3

where n1, n2 and n3 are integer numbers (…-2,-1,0,1,2,...)

and the three vectors a1, a2 and a3 are called 'primitive

lattice vectors'. They span the lattice.

The choice of the lattice vectors is not unique for a given lattice, as the Fig. 1 shows.

One lattice point can contain one single atom or a group of atoms, called basis of the crystal lattice.

Fig 1. [1]

Basis: 1 atomBasis: 1 atom Basis: 4 atoms

2D Bravais lattices

In 2D there exist 5 Bravais lattices:

Square: a1 = a2, α = 90°

Rectangular primitive: a1 ≠ a2, α = 90°

Centered rectangular: a1 ≠ a2, α = 90°

Hexagonal: a1 = a2, α = 120°

Oblique: a1 ≠ a2, α ≠ 90°

a1 and a2 are the lengths of the primitive lattice

vectors.

Fig 2. [2]

3D Bravais lattices

1. Triclinic: a1 ≠ a2 ≠ a3, α ≠ β ≠ γ

2. Monoclinic: a1 ≠ a2 ≠ a3, α = γ = 90°≠ β

2a: primitive, 2b: base centered 3. Orthorhombic: a1 ≠ a2 ≠ a3, α = β = γ = 90°

3a: primitive, 3b: base centered, 3c: body centered, 3d: face centered4. Hexagonal: a1 = a2 ≠ a3, α = β = 90°, γ = 120°

5. Rhombohedral: a1 = a2 = a3, α = β = γ ≠ 90°

6. Tetragonal: a1 = a2 ≠ a3, α = β = γ = 90°

6a: primitive, 6b: body centered7. Cubic: a1 = a2 = a3, α = β = γ = 90°

7a: primitive, 7b: body centered, 7c: face centered

In 3D: 7 crystal systems with 14 Bravais lattices

Fig 3. [2]

Outline

I. Introduction

II. Bravais lattices

III. Band structure theory

IV. Space groups

V. Symmetry properties of crystal orbitals and vibrations

VI. Group-subgroup relationships

VII. References

Band Structure Theory. I. Hückel TheoryHückel matrix H

Hμμ = (μ|heff|μ) = α

Hμν = (μ|heff|ν) = β if μν are nn

= 0 else

H = α E + β T

E unit matrix, T topological matrixS overlap matrix, Sμν = (μ|ν) = δμν

εj = α + β λj

λj eigenvalues of T

Alternative names:HMO = Hückel Molecular Orbital method (chemistry)TB = Tight Binding model (physics)

Characteristics:> one electron model> effective one electron Hamiltonian, heff

> simplest form for n atoms with one orbital (1s or 2pz) each, χμ, μ=1,n

Hückel: 1D Linear Chain

Fig. 6.Band width: 2βГ, Г coordination number

Hückel: Ring

Fig. 12 [6]

1D Hückel systems

TX1,TX2

Band Structure Theory. II. k-space

TX4a,TX4b

Reciprocal Lattice

We are looking for values of k that yield plane waves with the peridiocity of a given Bravais lattice.

exp(iK(r+R)) = exp(iKr)

for all R in the Bravais lattice.Obiously, the K values satisfying this requirement are given by

exp(iKR)=1

They span a lattice in the k-space (or „reciprocal space“) that is called the „ reciprocal lattice“.

The three vectors b1,b2,b3 are the „primitive

lattice vectors“ in the k-space, as a1,a2,a3 are

the primitive lattice vectors in the direct space.

bi * aj = δ ij

Any vector in the k-space can be written as

k= k1 b1 + k2 b2 + k3 b3

Fig. 11 [1]

TX5

Band structure Theory. III. Bloch functions

Brillouin zones> The Bloch functions are the basis functions for the translation group T.> Since T is an Abelian group, all irreps are non-degenerate.> The components kx,ky,kz of the wave vector k are labels, quantum numbers for

the irreps of T.> For a crystal of length L and n0 translations (in each direction) with a lattice

constant a=L/n0 the values of kx,ky,kz are limited to

kx,ky,kz = 0, ±2π/n0a, ±4π/n0a, ±π/a

> These values are contained in the first Brillouin zone (BZ).> Wave vectors in extended Brilloin zones with k=k+Km where Km is a reciprocal

lattice vector, have the phase factor exp(i(k+Km)Rn) = exp(ikRn)

Rn being a lattice vector in the direct lattice, since exp(iKmRn) = 1 .

> It is always sufficient to study the first BZ.

Band Structure Theory. IV. HF theory for crystal orbitals

• •

• •

• •

• •

• •

• •

Cell number -2 -1 0 1 2 3

Direct space:

Local basis functions: |i;p>cell i; basis p

Matrix element for Fock operator F: <i;p | F | j;q>

Translational symmetry: <i;p | F | j;q> = < 0;p | F | j-i;q>

Reciprocal space:

Fourier transform: Fpq (k) = Σj exp(iktj) <0;p | F | j;q>

tj translation vector to cell j

Diagonalization of the Fock matrix for all values of k. No off-diagonal elements of F between different values of k.

2D example

Valence bands of a 2D layer of NiO(100) Fig 13. [21]

Outline

I. Introduction

II. Bravais lattices

III. Band structure theory

IV. Space groups

V. Symmetry properties of crystal orbitals and vibrations

VI. Group-subgroup relationships

VII. References

Molecules: Point Group Symmetry Operations

Operation Schoenflies Hermann-Mauguin (Chemistry) (Crystallography)

Identity E 1Rotation through 2π/n Cn n

Reflection in a plane σ m 'horizontal' plane σh n/m

'vertical' plane σv nm

two nonequivalent vertical planes nmm 'diagonal' plane σd

Inversion i -1 Rotation reflection σhCn Sn

Rotation inversion iCn -n

32 crystallographic point groups

32 crystallographic point groups

(continuation)

Fig5

Crystals: Space groups, formal approach

TX3, Ref. 4 Space group is a group

Space group elements

Translations: (invariant subgroup), for each direction x,y,z Abelian groups 1-dim irreps, kx,ky,kz

Point group elements: Subgroup?

Compound elements: Glide planes, screw axes

Fig 14. [4]

Glide plane: Translation parallel to a given plane + reflection in that plane.Screw axis: Translation along a given axis and rotation through this axis.

Space groups: Symbols, Names

Symbols for symmetry elements:

Most important: ≬ 2-fold rotation ▲ 3-fold rotation ◾ 4-fold rotation ● 6-fold rotation

Many more, see Appendix A.

Notations for space groups:

Examples: P4/m, Immm

P primitive I body centered F face centered A,B,C base centred R rhombohedral

numbers rotations m mirror planes

Details: Ref. 1,7

Point groups,

space groups:

Schoenflies and Hermann-Mauguin

names

Ref. 7

Detour: Direct Products of Groups

Be G a group with the subgroups G and H

G = (G1, G2,..., Gg), order g

H = (H1, H2,..., Hh), order h

We build the compound elements Gi Hj

Multiplication of the compound elements

( Gi Hj ) ( Gk Hl ) = Gi Hj Gk Hl = Gi Gk Hj Hl = ( Gi Gk) ( Hj Hl ) = Gi' Hj'

The compound elements form a group, if the elements of H commute with those of G.

„Direct product“ of the groups G and H, K = G×H, order g·h

Space groups

In general, translations and point group operations do not commute.

{ε|τ'} {α|τ} = {εα|ετ+τ'} = {α|τ+τ'}

{α|τ} {ε|τ'} = {αε|ατ'+τ} = {α|ατ'+τ}

Therefore, in general a space group is not a direct product of the translational group T and a point group.

> Symmorphic space groups: They contain the entire point group as a subgroup. They are (semi)-direct products of the translation and point subgroups. They do not contain glide planes and screw axes.> Nonsymmorphic space groups: No direct products, contain glide planes and screw axes.

There exist 230 different space groups, 73 of them are symmorphic. Ref. 12.

2D space groups

For the five 2D Bravais lattices there are 17 2D space groups.

Their properties are given in the Table.

The full information is given in Ref. 12

2D space groups

Fig. 4 [4]

Outline

I. Introduction

II. Bravais lattices

III. Band structure theory

IV. Space groups

V. Symmetry properties of crystal orbitals and vibrations

VI. Group-subgroup relationships

VII. References

Group of k

We are interested in the symmetry properties of the Bloch funcions ψ(k,r) for different wave vectors k.

1. The point group of the reciprocal space is the same as for the direct space. (Proof?)

2. Definition: The group of (the wave vector) k is the set of space group elements that transform k into itself or into an equivalent k.

3. Definition: The star of k is the set of all wave vectors k' which are obtained by applying the point group elements on k.

Example 1: 2D square lattice

Space group: p4mmPoint group: (D4h) C4v

Lattice constant: a

Reciprocal space

2D square lattice: Space groups

p4C4

p4mmC4v

p4gm

Fig. 15 [4]

Star of k for the 2D square lattice

Г

Z

Δ

G

Σ

M

X

2D square lattice: High symmetry pointsSpace group: p4mm (symmorphic)Point group: C4v

Elements: E, 2C4,C2,2σv,2σd

1. Brillouin zone

k point type group of k

Г high symmetry C4v

M high symmetry C4v

X high symmetry C2v

Δ symmetry line {E,σv}

Σ symmetry line {E,σd}

Z symmetry line {E,σv}

G general {E}

Fig. 17 [5]

Irreps of the groups of k

2D square: k-space, reciprocal latticeFig. 9 [6]

s orbitals: Hückel theory

Hückel band structure of a squareFig. 7b [5]ε(kx,ky) =α +2β cos(kxa) + 2β cos(kya)

Node structure in the 2D planar Hückel system (s orbitals)

Fig. 10 [6]

Symmetry of the crystal orbitals at different k points

k point p4mm C4v

Γ Γ1 a1

M M4 b2

X X3 b2

Δ Δ1 a'

Σ Σ2 a''

Z Z2 a''

Symmetries at the high symmetry points

a1

b2

b2

a''a''

a'

px, py orbitals: Hückel theory

px orbitals: εx(kx,ky) =α -2βσ cos(kxa) + 2βπ cos(kya)

py orbitals: εy(kx,ky) =α -2βσ cos(kya) + 2βπ cos(kxa)

βσ

βπ

Fig. 18 [5]

Nearest neighbour interactions of py orbitals

Symmetry of the crystal orbitals at different k points

k point p4mm C4v

Γ Γ5 e

M M5 e

X X3+X4 b1+b2

Δ Δ1+Δ2 a'+a''

Σ Σ1+Σ2 a'+a''

Z Z1+Z2 a'+a''

The formulas show that the energy levels are degenerate at the points Γ and M:

Γ: ε = α - 2βσ + 2βπ M: ε = α + 2βσ – 2βπ

but not at the other points.

At the high symmetry points Γ and M they span the two-dimensional irrep Γ5 (or e), which is split at all other k points.

Again:NiO(100)

Valence bands of a 2D layer of NiO(100) Fig 13. [21]

Example 2: 2D hexagonal structures

graphene, p6mm ??, p6mm

2D hexagonal space groups

Number Space group

Subgroup of C6v

Point group elements Symmorphic

13 p3 C3 E, 2C3 yes

14 p3m1 C3v E, 2C3, 3σv yes

15 p31m C3d E, 2C3, 3σd yes

16 p6 C6 E, 2C6, 2C3, C2 yes

17 p6mm full C6v E, 2C6, 2C3, C2, 3σv, 3σd yes

Graphene

direct space reciprocal space

Nobel price in physics 2010 for K. S. Novoselov and A. K. Geim

Two trigonal sublattices:- Blue and yellow points in the direct space- Inequivalent points K and K' in the reciprocal space.

Lattice vectors:Direct space: a1 = a/2 (3,√3), a2 = a/2 (3, -√3)

Reciprocal space: b1 = 2π/3a (1,√3), b2 = 2π/3a (1, -√3)

a=1.42 Å is the C-C distance

Ref. 22

Hückel theory of graphene

Hückel band structure of grapheneFig. 8 [5]

Closed solution possible (Wallace 1947)

> + sign: lower (π) band> - sign: upper (π*) band> symmetric around α> high symmetry points: Γ: ε = α ± 3β M: ε = α ± β K: ε = α> Fermi energy (K point): EF = α

TB theory of graphene

Ref. 22> nn interaction: t = β= 2.7eV> nnn interaction: t' = 0.2 t> E(k) in eV

ε(k)Density of states (DOS)

Band structure of graphene

E. Kogan [24]

Hückel: Only s orbitalsFP-LAPW: All valence orbitals

Determination of the symmetries of crystal orbitals and vibrations

Reducible representation for a given k point (explicit treatment in Ref. 4) (for symmorphic space groups) Γ(k) = Γequiv(k) × Γvec

where Γequiv(k) is the representation of the lattice and Γvec the representation of

the orbitals or coordinates (vibrations).

The characters of Γequiv(k) are 1 for all elements of the group of k that transform an atom of the lattice into itself or into an equivalent atom and are 0 else.For k ≠ 0 the appropriate phase factors have to be included.

The usual rules are applied to determine the irreps which are contained in Γ(k).

For nonsymmorphic space groups everything is more difficult. Details in Ref. 4.

Crystal orbitals of graphene

k point Group of k Гequiv Гvec IrrepsГequiv × Гvec

Γ D6h a1g + b1u a1g+a2u+e1u A1g+A2u+E1u +B1u+B2g+E2g

K D3h e' a1'+a2''+e' E'+E''+A1'+A2'+E'

M D2h b1u+b2g ag+b1u+b2u+b3u Ag+Au+2B2g+B3g+2B1u+B3u

Гequiv : The two nonequivalent C atomsГvec

: The four valence orbitals at the C atoms: s,px,py,pz

2*4 = 8 crystal orbitals

Ref. 25

Vibrational modes of graphene

k point Group of k Гequiv Гvec IrrepsГequiv × Гvec

Γ D6h a1g + b1u (a2u)+e1u (A2u)+E1u +(B2g)+E2g

K D3h e' (a2'')+e' (E'')+A1'+A2'+E'

M D2h b1u+b2g (b1u)+b2u+b3u (Ag)+Au+B2g+B3g+B1u+(B3u)

Гequiv : The two nonequivalent C atomsГvec

: The three Carterian displacements coordinates x,y,z

2*3 = 6 vibrational modes (two out-of-plane modes, in parentheses)

Vibrational modes of graphene at the K point

Ref. 4

In-plane modes

Ref. 4

out-of plane modes

Graphene: Massless Dirac particles (Dirac cone)

For electrons with k in the vicinity of the K point,i.e. k = K+q and |q| ≤ |K| we have: E(q) = vF |q| +O ( (q/K)2 )

vF = 3βa/2

is the Fermi velocity. It does not depend on the energy E.

Usually, one has for a particle with mass m E(q) = q2 /2m v = q/m = √(2E/m)Here, v changes with the energy.

Classical relativistic movement:E2 / c2 = p2 + m² c2

2D Dirac equation:(-i vF σ ∙∇ - m ) ψ(r) = E ψ(r)

Properties of graphene

> Half metal> Insulator with zero band gap> Pseudorelativistic electrons at the Fermi energy> Anomalous Quantum Hall effect > High electron and hole mobility at room temperature> Small resistivity, even less than silver> Large mechanical and chemical stability

> Several chemical functionalizations possible > Sensors, solar cells, ...> Li coated graphene exhibits superconductivity> Quantum computers (??)> ...

← Physics

← Applications

Outline

I. Introduction

II. Bravais lattices

III. Band structure theory

IV. Space groups

V. Symmetry properties of crystal orbitals and vibrations

VI. Group-subgroup relationships

VII. References

Group-subgroup relationships

Reduction of symmetry:

> Change of atoms > Contaminations> Jahn-Teller systems> Peierls distortion> …> Temperature effects; phase transitions

Symmetry of the crystal is loweredSpace group is replaced by a (smaller) subgroup

Subgroups of space groups

Pm-3m; point group Oh

Pbcm; point group D2h

Ref. 7

Bärnighausen Tree, example 1

Ref. 7

From diamond to zinc blende:

> 'translationsgleich'> point groups diamond: Td

zinc blende: Td

> Wyckoff positions: diamond: 8a zinc blende: 4a,4c

Ref. 25

Bärnighausen Tree, example 2

Boron compunds

> translationsgleich> klassengleich> isomorphic

Peierls distortion

Peierls distortion (physics)Bond alternation (chemistry)

Example: Linear chain, alternant distances, long and short bondsHückel theory: β1, β2

1. Brillouin zone: ε(0) = α ± (β1+β2)

band gap: W = 2 (β1 – β2 )

Linear chain with 35 H atomsHückel theory: α = 0 β2 = 0.75 β1

Peierls distortionLinear chainHückel theoryLattice constant: a=1

One atom in unit cell:0 ≤ k ≤ π/a

Two atoms in unit cell:0 ≤ k ≤ π/2a

β2 = 0.75 β1β2 = β1

Jahn-Teller TheoremJahn-Teller Theorem:

If a (nonlinear) molecule possesses a (spatially) degenerate electronic state, there exists always a vibrational mode (distortion of the nuclear frame) that lifts the degeneracy.

Proofs:

1. H. A. Jahn, E. Teller, Proc. R. Soc. London A161, 220 (1937) By checking all possible point groups.2. E. Ruch, A. Schönhofer, Theoret. Chim. Acta 3, 291 (1965) By using group theoretical methods.

Group theoretical formulation:

If the electronic state belongs to a multi-dimensional irrep Γ, then the (nontotally symmetric) vibrations that are contained in [Γ2] will lift the degeneracy.

Jahn-Teller splitting: Example

Let px and py be two orbitals in an environment

of D4h symmetry.

The pair px,py spans the irrep Γ = eu of D4h.

The direct product of Γ with itself is Γ2 = Γ × Γ = a1g + {a2g} + b1g + b2g

{a2g} is the antisymmetric part of the product.

Therefore the two vibrational modes b1g and

b2g split the irrep eu. Jahn-Teller splitting: The energy levels W+ and W- as

functions of one mode Q .

Alkali Hyperoxides, AO2, A=K,Rb,Cs

Ref. 27 Ref. 28

Rubidium superoxide, RbO2

Crystal structure: tetragonal a=4.24 Å, c=7.03 Å below 15K: weakly monoclinic Molecular unit: Rb+O2

- R(O2

-)=1.350 ÅAntiferromagnetic Néel temperature: 15 K Curie-Weiss temperature: -26 KElectronic states: Rb+: 1S O2

-: 2Π, SOC: 2Π1/2 , 2Π3/2, 160 cm-1

Electronic structure, O2-

3σu

1πg

1πu

3σg

Molecular orbitals,derived from O2p

Electronic states

O2

..1πu

41πg2 3Σ

g-

O2

- ..1πu41π

g3 2Π

g

O2

2- ..1πu41π

g4 1Σ

g+

Spin-orbit coupling (SOC) in O2-:

2Π → 2Π1/2

, 2Π3/2

Splitting of 160 cm-1 = 20 meV

Embedded cluster approach

Cluster setup for the interaction of two O2- anions

in RbO2

O2

- + O2

- + 10 Rb+ + 4 ECP +PCF (3404)

each O2- must be fully surrounded by Rb+ ions

ECPs for O2

- (not necessary)

Jahn-Teller Splitting in RbO2One O2

- molecular ion in a D4h environment:

Electronic ground state is degenerate without SOC: 2Π (twofold space and spin) with SOC: Π

3/2 (twofold spin)

Questions: 1. Which vibrational mode removes the degeneracy? 2. How do SOC and Jahn-Teller splitting interact? 3. Comparison with experiment?

Cluster claculations: > One O2- in the cluster

> Crystal structure preserved > One vibrational mode activated

B1g mode

B2g mode

Magnetic coupling paths

parallelJ = -7.90Dzz=-14.04

skewJ = -18.71Dzz= -21.28

diagonalJ = -1.11Dzz= -0.99

linearJ = -0.65Dzz= +0.01

All values in cm-1; F. Uhl, V. St., unpublished

Outline

I. Introduction

II. Bravais lattices

III. Band structure theory

IV. Space groups

V. Symmetry properties of crystal orbitals and vibrations

VI. Group-subgroup relationships

VII. References

Textbooks

1. N. W. Ashcroft, N. D. Mermin, Solid State Physics, Saunders College Publishing, 19762. A. Zangwill, Physics at Surfaces, Cambridge University Press, 19883. K. Kopitzki, Einführung in die Festkörperphysik, Teubner Taschenbücher, 19894. M. S. Dresselhaus, G. Dresselhaus, A. Jorio, Group Theory. Application to the Physics of Condensed Matter, Springer-Verlag, 20105. J. Y. Burdett, Chemical Bonding in Solids, Oxford University Press, 19956. R. Hoffmann, Solids and Surfaces: A Chemist's View of Bonding in Extended Systems, VCH Publishers, 19887. U. Müller, Symmetry Relationships between Crystal Structures, Oxford University Press, 20138. S. L. Altmann, Band Theory od Solids: An Introduction from the Point of View of Symmetry, Clarendon Press, Oxford, 1991

Tables

11. S. L. Altmann, P. Herzig, Point-Group Theory Tables, Clarendon Press, Oxford 199412. International Tables for X-ray Crystallography13. R. W. G. Wyckoff, Crystal Structures, Interscience Publishers, 1965

Papers

21. A. Leitheußer, Dissertation, Bochum 200122. P. R. Wallace, Phys. Rev. 71, 622 (1947)23. A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, A. K. Geim, Rev. Mod. Phys. 81, 109 (2009)24. E. Kogan, Graphene 2, 74 (2013)25. E. Kogan, V. U. Nazarov, V. M. Silkin, E. E. Krassovskii, arXiv:1306.4084v326. R. X. Fischer, H. Schneider, Eur. J. Mineral. 20, 917 (2008)27. M. Labhart, D. Raoux, W. Känzig, M. A. Bösch, Phys. Rev. B 20, 53 (1979)28. W. Hesse, M. Jansen, W. Schnick, Prog. Solid St. Chem. 19, 47 (1989)29. F. Uhl, V. Staemmler, unpublished

Appendix A

Ref. 7, p78

Appendix B1

Character Table D3h

Ref. 11

Appendix B2

Character Table D6h

Ref. 11